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1: 1.4 Calculus of One Variable

… ►If $f(x_1)\le f(x_2)$ for every pair $x_1$, $x_2$ in an interval $I$ such that , then $f(x)$ is nondecreasing on $I$. … ►If also $f(x)$ is continuous on the right at $x=a$, and continuous on the left at $x=b$, then $f(x)$ is continuous on the interval $[a,b]$, and we write $f(x)\in C[a,b]$. … ►If $f(x)$ is continuous on an interval $I$ save for a finite number of simple discontinuities, then $f(x)$ is piecewise (or sectionally) continuous on $I$. … ►If $f^{(n)}$ exists and is continuous on an interval $I$, then we write $f\in C^n(I)$. …

2: 26.2 Basic Definitions

… ►Unless otherwise specified, it consists of horizontal segments corresponding to the vector $(1,0)$ and vertical segments corresponding to the vector $(0,1)$. For an example see Figure 26.9.2. … ►A partition of a set $S$ is an unordered collection of pairwise disjoint nonempty sets whose union is $S$. … ►A partition of a nonnegative integer $n$ is an unordered collection of positive integers whose sum is $n$. As an example, $\{1,1,1,2,4,4\}$ is a partition of 13. …

3: 28.17 Stability as $x\to \pm \mathrm{\infty }$

… ►However, if $\mathrm{\Im }\nu \ne 0$, then $(a,q)$ always comprises an unstable pair. …

4: 18.36 Miscellaneous Polynomials

… ►These are OP’s on the interval $(-1,1)$ with respect to an orthogonality measure obtained by adding constant multiples of “Dirac delta weights” at $-1$ and $1$ to the weight function for the Jacobi polynomials. …

5: Bibliography K

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Algorithm 763: INTERVAL_ARITHMETIC: A Fortran 90 module for an interval data type. ACM Trans. Math. Software 22 (4), pp. 385–392.

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ISSN 0098-3500, Document, GAMS (module 763; package TOMS), ZentralBlatt

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6: 18.2 General Orthogonal Polynomials

… ►It is assumed throughout this chapter that for each polynomial $p_n(x)$ that is orthogonal on an open interval $(a,b)$ the variable $x$ is confined to the closure of $(a,b)$ unless indicated otherwise. (However, under appropriate conditions almost all equations given in the chapter can be continued analytically to various complex values of the variables.) … ►More generally than (18.2.1)–(18.2.3), $w(x)dx$ may be replaced in (18.2.1) by a positive measure $d\alpha (x)$, where $\alpha (x)$ is a bounded nondecreasing function on the closure of $(a,b)$ with an infinite number of points of increase, and such that for all $n$. … ►All $n$ zeros of an OP $p_n(x)$ are simple, and they are located in the interval of orthogonality $(a,b)$. …

7: 10.23 Sums

… ►provided that $f(t)$ is of bounded variation (§1.4(v)) on an interval $[a,b]$ with . …

8: 1.6 Vectors and Vector-Valued Functions

… ► $𝐜(t)=(x(t),y(t),z(t))$, with $t$ ranging over an interval and $x(t),y(t),z(t)$ differentiable, defines a path. … ►with $(u,v)\in D$, an open set in the plane. …

9: 2.3 Integrals of a Real Variable

… ► $k$ ($\le \mathrm{\infty }$) and $\lambda$ are positive constants, $\alpha$ is a variable parameter in an interval ${\alpha }_1\le \alpha \le {\alpha }_2$ with ${\alpha }_1\le 0$ and , and $x$ is a large positive parameter. …

10: 18.1 Notation

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$x,y$	real variables.
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$w(x)$	weight function $(\ge 0)$ on an open interval $(a,b)$.
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